An alternate view of complexity in k-SAT problems

The satisfiability threshold for constraint satisfaction problems is that value of the ratio of constraints (or clauses) to variables, above which the probability that a random instance of the problem has a solution is zero in the large system limit. Two different approaches to obtaining this threshold have been discussed in the literature - using first or second-moment methods which give rigorous bounds or using the non-rigorous but powerful replica-symmetry breaking (RSB) approach, which gives very accurate predictions on random graphs. In this paper, we lay out a different route to obtaining this threshold on a Bethe lattice. We need make no assumptions about the solution-space structure, a key assumption in the RSB approach. Despite this, our expressions and threshold values exactly match the best predictions of the cavity method under the 1-RSB assumption. Our method hence provides alternate interpretations as well as motivations for the key equations in the RSB approach.Comment: 5 pages, 3 figures, typos correcte

On the behaviour of random K-SAT on trees

We consider the K-satisfiability problem on a regular d-ary rooted tree. For this model, we demonstrate how we can calculate in closed form, the moments of the total number of solutions as a function of d and K, where the average is over all realizations, for a fixed assignment of the surface variables. We find that different moments pick out different 'critical' values of d, below which they diverge as the total number of variables on the tree goes to infinity and above which they decay. We show that K-SAT on the random graph also behaves similarly. We also calculate exactly the fraction of instances that have solutions for all K. On the tree, this quantity decays to 0 (as the number of variables increases) for any d>1. However the recursion relations for this quantity have a non-trivial fixed-point solution which indicates the existence of a different transition in the interior of an infinite rooted tree.Comment: 22 pages, 5 figures,accepted for publication in J. Stat. Mec

A PP-Adic Spectral Triple

We construct a spectral triple for the C$^*$-algebra of continuous functions on the space of $p$-adic integers by using a rooted tree obtained from coarse-grained approximation of the space, and the forward derivative on the tree. Additionally, we verify that our spectral triple satisfies the properties of a compact spectral metric space, and we show that the metric on the space of $p$-adic integers induced by the spectral triple is equivalent to the usual $p$-adic metric

A Stochastic model for dynamics of FtsZ filaments and the formation of Z-ring

Understanding the mechanisms responsible for the formation and growth of FtsZ polymers and their subsequent formation of the $Z$-ring is important for gaining insight into the cell division in prokaryotic cells. In this work, we present a minimal stochastic model that qualitatively reproduces {\it in vitro} observations of polymerization, formation of dynamic contractile ring that is stable for a long time and depolymerization shown by FtsZ polymer filaments. In this stochastic model, we explore different mechanisms for ring breaking and hydrolysis. In addition to hydrolysis, which is known to regulate the dynamics of other tubulin polymers like microtubules, we find that the presence of the ring allows for an additional mechanism for regulating the dynamics of FtsZ polymers. Ring breaking dynamics in the presence of hydrolysis naturally induce rescue and catastrophe events in this model irrespective of the mechanism of hydrolysis.Comment: Replaced with published versio